FUNCTIONS: AN APPROACH USING COOPERATIVE LEARNING AND TECHNOLOGY.
ANTICIPATING THE YEAR 2000
UNDERSTANDING OF PHYSICAL MOVEMENTS WITH PSL

"It is the consensus of virtually all the men an women who
have been working on curriculum projects that making material interesting
is in no way incompatible with presenting it soundly; indeed, a correct
general explanation is often the most interesting of all."
(Jerome Bruner, The Process of Education, 1960, p 23.)

AbstractIn this paper we offer those teachers who want to enhance their teaching
of mathematics with science one class that uses a PSL in the exploration
of the relationship between distance and time in two particular situations,
in an environment that supports cooperative learning.
The paper is organized in three main sections. First section presents the
relevance of this project inside our discipline. The second section presents
the class activity that we have designed; this section is divided in four
subsections: Objectives, Methodology, Mathematical Content and Assessment;
the last section presents our conclusions, and suggestions for further improvement
of the class. We include five appendix with some detailed information about
PSL, two worksheets of PSL activities, an evaluation check list, and pair
test.
We think that the use of technology will make a significant contribution
to teaching of mathematics. We encourage teachers to get a chance to see
that the new generation of teaching of mathematics with technology is more
than a dream.

Introduction

Why this topic?
According to Kieran (1992, pp. 408) the topic of Functions has been presented
to the students from a structural perspective &shyp;using the set theory
definition&shyp; and using principally the symbolic representation and promoting
translations from the symbolic to the graphical representation. As a consequence:
· The constant function, the piece-wise function and the function
represented as a discrete set of points are the ones that causes more difficulties
among students.
· Students neglect domain and range.
· Students do not understand completely the concept and representation
of images and pre-images, both in symbolic and in graphical representations.
· The variety of examples students have is limited; with a high preference
of linearity.
· Students find easier to translate from symbolic to graphic representation,
than from graphic to symbolic representation.
Nevertheless, the development of the hand held technology and of microcomputers'
software, has changed the perspectives for teaching functions in the school.
According to the standards, today is a requirement that students:
· understand the concepts of variable, expression and equation;
· represent situations and number patterns with tables, graphs, verbal
rules, and equations and explore the interrelationships of these representations;
· represent situations that involve variable quantities with expressions,
equations, inequalities, and matrices;
· use tables and graphs as tools to interpret expressions, equations,
and inequalities (National Council of Teachers of Mathematics, 1989, p.
102, p. 150).
So the concept of function seen as an integrator in the curriculum can be
taught in a meaningful way.
In the society of twentieth century, people needs cooperative skills more
to be successful in their jobs and in their daily lives. The significance
of cooperative learning lies in the fact that this method provides a social
environment where people freely can exchange ideas, ask questions, help
one another to understand the ideas in a meaningful way, and to express
what they feel and think. On the other hand, research on cooperative learning
indicates that in order to become successful and confident problem solvers
and in order to work on mathematics projects, students need to work cooperatively
(Johnson and Johnson, 1990).
In this activity, our objective using cooperative learning technique is
to help students gain these skills, and to help them learn how to use these
skills. They will learn and practice basic characteristics of cooperative
learning when they are working as a team to attain their common goal.There
are two explicit reasons for using this methodology in our class.
Learning by different ways.
People learn by talking, listening explaining, and thinking with others,
as well as by tehmselves. In addition to that a test of understanding is
often the ability to communicate to others (that is also what we have done
in oral type or essay type of examinations assessing the students knowledge)
and this communication act is itself often the final and most crucial step
in the learning process.
Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989)
addresses the same issue:
Small group work, large-group discussions, and presentation of individual
and group reports -both written and oral- provide an environment in which
students can practice and refine their growing ability to communicate mathematical
thought processes and strategies.
Small groups provide a forum for asking questions, discussing ideas, making
mistakes, learning to listen to other's ideas, offering constructive criticism,
and summarizing discoveries in writing. (p.79)
Handling richer and more challenging situations.
In mathematics, there can be created many opportunities for creative thinking,
for exploring open-ended situations, for making conjectures and testing
them with data, for posing intriguing problems, for solving non-routine
problems.
Students in groups can often handle challenging situations that are well
beyond the capabilities of the individuals at the developmental stage. As
an individual attempting to explore those same situations, students in general
often make little progress and experience unnecessary frustration (Davidson,1980).

Class Design
We are going to present our class design talking about its four main components:
Objectives, methodology, content, and assessment (Rico, 1991).

Objectives
Students will appreciate the value of mathematical representation by deriving
and understanding corresponding mathematical representation for a physical
event.
Students will be able to predict and verify the corresponding physical structure
for a given mathematical representation.
Students will be able to explain the relationship between motion and mathematical
representation.
Students will be able to use a graph for describing a motion and communicate
their mathematical thinking.
Students will be able to mathematize a given physical event and give reasoning
to their representation.

Methodology
Students will use the PSL for understanding physical motion. We assume that
students are familiarized with coorperative learning environment since the
activities will be performed groups of three or four members.
The followings are the descriptions of the two tasks proposed. The idea
is that more than one group work on each task, so at the end, the class
will be able to compare different approaches and results. Activity 1 comes
from the PSL Teacher's Edtion Book (IBM, 1990, p.1-18). The goal is to find
both, the physical explanation and the symbolic representation from a given
graph that represents a motion. The goal of Activity 2 is to produce the
graph that corresponds to a description of a motion and to resepresent symbolically
the graph.

Activity 1: A Challenge for You Cats.
The educational environment is much like that of CBL (Calculator Based
Laboratory). A cat-like graph is given to students (IBM, 1990, p. 1-18).
Observing the graph is the initial and fundamental thinking process by which
information is acquired.
The aim of the activity is to find and verify mathematical representation
when the physical representation is given. Students are expected to improve
their intuition about the phenomena being studied and mathematical knowledge
being involved. This activity has a potential since many different concepts
of functions studied in mathematics class should be used in doing PSL activity.
See Appendix 2 for the detailed text of the Activity 1.
After studying the graph, students open a copy of the graph on PSL's screen.
When they describe the movement that would produce the graph each student
can compare and relate the similarities and differences of his or her thought
with the other students. This experience will provide the students with
the opportunity for active participation. They will talk about the speed
, time, and direction of the movement one another within a group. The application
of physical science to mathematics is truly realized since the importance
of the quantitative nature of science is stressed out in the exploration
in many ways. Students need to be very sensitive to the speed, time, and
movement that they do within the experiment. Changing the range of values
on the axes is also understood conceptually when they want to have a better
fitted graph because students are taking care of real data from their activity.

Activity 2: Trace the Target.
The second activity is reversed form of the first activity. We ask students
to graph a given physical movement description.
The aim of the activity is to find the mathematical representation of a
described motion. Students will be able to understand and analyze the corresponding
movements required for a shown graph. Students will be able to mathematize
the movement and construct the corresponding piece-wise functions. See Appendix
2 for the detailed text of the activity.
After studying the description, students work on the movement that will
correspond to the given graph. Then student should create the corresponding
graph on the computer's screen. After analysing the graph that would produce
the movement described, each student can compare and relate the similarities
and differences of his or her thought with the other students. Such an experience
will foster active participation and mathematical communication for each
student. They will talk about the speed, time, and direction of the movement
one another within a group. The application of physical science to mathematics
is truly realized since the importance of the quantitative nature of science
is stressed out in the exploration in many ways. Students need to be very
sensitive to the speed, time, and movement that they do within the experiment.
Approximation and graph fitting will also be understood conceptually, because
during experiment they will be trying to fit to their predicted equation
and they will also be modifying their graph if necessary.
Every student will have a PSL program built in an IBM computer, PSL equipment,
a worksheet, and a check-list for cooperative work asessment. The use of
graphing calculators are allowed when the students feel the need of them
to find a graph. Time for the group work is 30 minutes. After the group
investigation students should present to the other groups their result.
A teacher will act as a facilitator of students' active exploration. The
teacher have to explain to the students the main goal of the activities.
He or she will monitor the students' activity and make a note for further
feedback to the students.

Content
We have two categories for the content of the PSL activities, the mathematical
and the science content.
The mathematical content covered by these two activities includes: Linear,
quadratic, and constant functions; Cartesian graphs, meaning of x and f(x);
piece-wise functions; domain and range of a function; slope of a line; maximum
of a parabola and linear equations. Further, this mathematical content will
include system of linear equations, symbolic forms of linear, quadratic,
and constant functions, graphs of linear, quadratic, and constant, translation
from symbolic to graphical representation and vice versa, mathematical models
for verbal situations, and description of mathematical situations.
The science content includes movement, speed and time, measurements, and
their mutual relationships.

Assessment
We have designed five different assessment tools for group and individual
achievement, attitude, and performance. We want to emphasize that these
tools are designed to provide the teacher information on those aspects,
but that the final grade should be the same for every student in the group,
since this is one of the main characteristics of the cooperative learning
strategy. This implies that the teacher has to establish a percentage of
each part, the individual component and the group component, in such a way
that that final grade assigned to the group reflects not only the group
achievement but each student's achievement as well.
We have designed five tools for assessing these components, Table 1 shows
the name of each tool with a brief description. Table 2 shows how the tools
can be used for assessing the components.
Table 1: Description of the assessment tools.
Name of the tool Description
Answer Sheet (AS) This sheet has the answers of the group to the worksheet,
and a group reflection about the activity. (See Appendix 2)
Oral Presentation (OP) This corresponds to the presentation that one person
of the group, selected by the group, presents the results of the exploration.
Each group has 5 minutes and one transparency. They have to decide what
to present. In a further discussion, teacher may ask students the rationale
for choosing the topics of the presentation.
Cooperative Work Checklist (CL) This sheet contains the opinion of each
student about the group and individual performance as a group in the particular
task. Teachers can fill their own checklist for each group, according to
their perception of the groups' work. (See Appendix 3)
Reflection (RF) This corresponds to the writing that each person in the
group produces after the activity. In it students will explain how the activity
helped them to gain a better understanding of mathematics and science concepts
as well as comments for improving the activity and the group work.
Pair Test (PT) This corresponds to the answer to the open question designed
for assessing the objectives of the class. Students will work in pairs during
50 minutes producing a two page paper that shows how they arrived to the
solution and what are the connections to real life. They have to show a
deep understanding of the concepts and a good communication strategy. (See
Appendix 4)
Table 2: Use of the tools for assessing each component.
Components Individual Group
Achievement RF, PT AS, OP
Performance CL AS, OP, CL
Attitude RF CL
The purpose of this last table is to show teachers how to use each of the
tools for obtaining information about the different aspects covered in the
task. Observe that self-assessment is considered when students answer the
checklist. We agree with the Assessment Standards For School Mathematics
(NCTM, 1995, p. 2) in the importance of having more than one source of information
for assessing our students. Hence the five tools proposed.

Conclusions and Suggestions
One thing which is really great for students is that mistakes are allowed.
Students don't have be discouraged by mistakes. If their movement doesn't
fit to the graph the PSL equipment and PSL software make it easy for them
to repeat an experiment. Students can try again and again until the result
is satisfactory.
Communication skills are very important when they express their ideas and
insights to write the movement that they need to produce the graph. Writing
skill is also required when they describe the movements for the graph. The
application of the concept or information gained in the exploration should
be combined with their mathematical knowledge of functions when they try
to find a piece-wise function which can explain the graph in symbolic mathematical
representation.
A graphic calculator would be allowed in finding the functions. Students
will be able to use the coordinate system if they want to find an exact
form of piece-wise functions. Students are expected to have the higher-level
of ability in applying and analyzing the graphs. Students will be able to
appreciate the effectiveness of a scientific activity in learning of mathematical
concepts.
There are several ways to implement cooperative learning method in classroom.
Among the possible cooperative activities that can be used in mathematics
classroom, the following list presents those characteristics that define
a good cooperative task:
one that will suit all ability levels,
one that needs many contributions,
one that involves manipulatives,
one that involves challenging activities,
one that involves pencil and paper activities,
one that involves games,
one that includes experiments.
These type of cooperative tasks help the process of cooperative learning
and thereby, provide a perfect atmosphere and a social context to improve
mathematical communication and to study mathematics.
Doing this project we have discovered that Bruner was true when he said
that
" making material interesting is in no way incompatible with presenting
it soundly; indeed, a correct general explanation is often the most interesting
of all."
The access to different technologies in our classrooms together with a
positive attitude towards change will make possible a new generation of
mathematics teaching. We are in a privileged position as practitioners and
as researchers, since we can see the two sides of the same coin: we can
make interesting material, but not only can present it soundly, but in a
way that is, we hope, really profitable to our students. We hope that this
contribution will give many teachers the possibility of taking the risk
of doing new things with a high probability of success: teachers and students
not only will have fun but will begin to find new ways to go across the
school curriculum in an easier way. We encourage the teachers that have
read this contribution to give us a chance to see that this dream, the new
generation of teaching of mathematics with technology, is more than a dream.

tetrahedra for the curriculum. Curriculum design, development, and evaluation].
Unpublished

document. Granada: Universidad de Granada.

APPENDIX 1: The PSL
General Description
The Personal Science Laboratory, PSL is a microcomputer-based laboratory
(MBL). This means that it allows students to collect data in situ,
to manipulate variables, to analyze data that can be presented in both graphical
and tabular modes, and to see immediate results from one experimental procedure.
It was released by the IBM in 1990.
The PSL operate on any IBM PS/2 with monochrome or color display and graphics
capability or on any IBM compatible, &shyp;286 or higher&shyp; with a 3.5"
or 5.25" diskette drive, a graphics display and an asynchronous adapter
card with a 25-pin connector (a serial port). It supports printer and fixed
disk drive. The PSL requires DOS 3.0 or above.
Besides the publications that support the operation of the PSL &shyp;the
PSL Hardware User's Guide and the PSL Technical Reference&shyp;, there are
three publications that are of pedagogical interest:
· PSL Student Version, contains instructions for running experiments
and raise questions that provide learning opportunities to the students.
· PSL Annotated Teacher Version contains the answers to the questions
posed to the students and the result of the completed experiments.
· PSL Student Versions Blackline Master Edition, offers worksheets
that can be freely photocopied and distributed in the school that uses the
PSL.
Components of the PSL
Roughly speaking a PSL is composed of one base unit to which four modules
can be connected. There are two types of modules the Motion and Mechanics
and the TLp (Temperature, Light, and pH) modules. The first is used to connect
the Motion and Distance probe, and the second to connect the Temperature,
Light, and pH probes (one jack for each). As an interesting feature Temperature
and Light probes can be used in any of the jacks of the TLp module; this
gives the possibility of collecting data from 12 different sources, for
example from 12 different students, with only one base unit.
The PSL is controlled by a very powerful software, the 'PSL Explorer'. The
software offers the following functions:
· to create, run, and retrieve experiments
· to analyze data of one experiment
· to export collected data to a spreadsheet
· to import data from a spreadsheet
An experiment consists of a set of parameters and the data collected during
a defined period of time. The probes are used for sending the computer the
value of the measured variable (distance measured in meters, temperature
measured in Celsius, light intensity measured both as absolute or corrected
in a way similar to the human eye and pH in pH units) at specific intervals
of time, or distance, or volume. After running the experiment the software
presents a graph of the two variables used in a Cartesian plane. A table
of the data collected is produced too. The analysis option allows the manipulation
of both the graph and the table; it offers functions for altering the graph
&shyp;zoom, regression line drawing, change of coordinates&shyp; and for
altering the data, &shyp;add, subtract, multiply or divide by a constant,
perform log and antilog operations, compute power operations (by rational
numbers); take reciprocals, calculate sines and cosines and differentiate
and integrate the y-axis with respect to the x-axis.
Features
The PSL is an interactive tool that provides teachers a large amount of
possibilities for exploring scientific ideas in relationship with mathematics.
Contrary to what can be expected setting up the hardware and the software
is an easy task. The package offers a variety of pre-designed experiments
for using each of the probes. There are some concerns about its cost: It
may be unaffordable for small schools.

APPENDIX 2: Activity 1: A Challenge for You Cats
A. Objectives:
The aim of the activity is to find and verify mathematical representation
when the physical representation is given.
Students are expected to improve their intuition about the phenomena being
studied and mathematical knowledge being involved.
This activity has a potential since many different concepts of functions
studied in mathematics class should be used in doing PSL activity.
B. Directions:
a. Study the following graph, then use the Disk option to load the
target file TAE1C11.PSL and display a copy of the graph on PSL/s
screen.

b. Write a description of the movement that would produce the graph.
c. Use PSL to measure the movement. Sketch PSL's graph with individual data.
d. Write a description of the graph. i.e. find a piece-wise function for
the graph.
e. Discuss the difference with your group members if your result does not
match the target graph.
C. Conclusions

APPENDIX 3: Activity 2: Trace the Target
A. Objectives:
The aim of the activity is to find the mathematical representation of a
described motion.
Students will be able to understand and analyze the corresponding movements
required for a shown graph.
Students will be able to mathematize the movement and construct the corresponding
piece-wise functions.
B. Directions:
a. Read the following description of a movement:
Description of the Movement:
I am a turtle who stands 1.5 meters from the probe for 5 seconds. Then I
begin to walk with a decreasing speed for 5 seconds and reach three meters
zone form the probe. When I get there, at the tenth second, I jump 40 cm
backwards and then I jump 30 cm forwards. From the 11th seconds to 15th
I stay there, and then I jump backwards 30 cm and forwards 40 cm. Without
waiting, at the 16th second, I began to walk towards probe with an increasing
speed and I suddenly stop when I reach 1.5 meters zone at the 21st second.
In the rest time, I stay there.
b. Describe the graph
c. Think about the mathematical representation of the given graph. Discuss
the curves that will corresponds to the movement described above. Discuss
the piece-wise functions that corresponds to the described movement.
d. Corresponding piece-wise function (only for time interval 0-13 seconds.
e. Plot the corresponding graph.
f. Use PSL to perform the described movement and obtain the graph of your
movement.
g. Compare your graph you have found in PSL and the described graph below.
Discuss.
C. Conclusion:

APPENDIX 4: Evaluation Check List
The following items are designed to assess individual and group cooperative
work. Each member of the group should complete this assessment sheet by
the end of the activity and should submit it to the teacher. Give your response
to each item and mark the appropriate box. In the response scale, 6 denotes
excellent (Exc) and 0 denotes Not Applicable (NA).